Unitary Groups Acting on Grassmannians Associated with a Quadratic Extension of Fields

Abstract

Let (V, H) be an anisotropic Hermitian space of finite dimension over the algebraic closure of a real closed field K. We determine the orbits of the group of isometries of (V, H) in the set of K-subspaces of V . Throughout the paper K denotes a real closed field and K its algebraic closure. Then it is well known (see, for example, [4, Chapter 2], [23]; see also [8]) that K = K(i) with i = √−1. Also we let (V,H) be an anisotropic Hermitian space (with respect to the involution underlying the quadratic field extension K/K) of finite dimension n over K. In this context we consider the natural action of the unitary group U = U(V,H) of isometries of (V,H) on the set Xd of all ddimensional K-subspaces of V . The analogous problem where (V,H) is a symplectic space was treated in [1] (for arbitrary quadratic field extensions). It turns out that, in contrast with the symplectic case, there are infinitely many orbits for the action of the unitary group U on Xd. In group theoretic language the stated problem turns into the determination of the double coset spaces of the form (1) GW \G/U, where G = GL (VK) and GW denotes the parabolic subgroup of G stabilizing a member W ∈ Xd (we write VK to indicate that we are regarding V as a vector space over K). The precise structure of double coset spaces involving classical groups is of great interest in applying the classical Rankin-Selberg method for explicit construction of automorphic L-functions, as Garrett [2] and Piatetski-Shapiro and Rallis [6] worked out. 2000 AMS Mathematics Subject Classification. Primary 51N30, 15A21, Secondary 11E39. Received by the editors on October 13, 2003. Copyright c ©2006 Rocky Mountain Mathematics Consortium